Problem 7 x:=<u,v,u*v>;

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xu:=map(diff,x,u);xv:=map(diff,x,v);

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xu_cross_xv:=LinearAlgebra:-CrossProduct(xu,xv);

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N:=xv_cross_xv/sqrt(LinearAlgebra:-DotProduct(xu_cross_xv,xu_cross_xv)) assuming real;

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Sxu:=-simplify(map(diff,N,u)) assuming real;

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Sxv:=-simplify(map(diff,N,v)) assuming real;

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c:=(1+u^2+v^2)^(3/2);

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Sxu_Oprea:= simplify(-u*v/c*xu+(1+v^2)/c*xv);

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Sxv_Oprea:= simplify((1+u^2)/c*xu-(u*v)/c*xv);

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Verify that our action of the shape operator agrees with Oprea's: simplify(Sxu-Sxu_Oprea);

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simplify(Sxv-Sxv_Oprea);

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